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Given an arbitrary shape in a plane with uniform density, a perimeter of 100, and it contains the origin, what is the probability that the center of mass of the shape is contained inside a circle with radius 5 centered at the origin?

This question is a question my friend thought up of, but I'm stumped on how to start. I've considered constructing a formula for finding the number of shapes which fulfill the conditions given a center of mass, but it's been a daunting task. If it's any help, I've noted the trivial fact that the center of mass is always contained in a circular region with radius 50. Symmetry might be of some use as well, but I'm not too sure how I'd use it to simplify the problem in a meaningful way.

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    $\begingroup$ I can't even envision a canonical way of choosing a random shape meeting the constraint. Since the answer will depend on the distribution of the random shape being used, you have to specify this. $\endgroup$ Commented Mar 14, 2023 at 6:37
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    $\begingroup$ This depends a lot on what you mean by "shape," and by the perimeter of an arbitrary shape. I imagine you need to add a lot of restrictions for this problem to be possible. As @SangchulLee said, you also need to then specify the distribution of these shapes. I don't think you will find a quick answer for this, but there may be a similar problem in this spirit which is. $\endgroup$
    – pancini
    Commented Mar 14, 2023 at 6:40
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    $\begingroup$ To further elaborate my point, note that even a naïve notion of 'random triangle' quickly becomes problematic as we see in Bertrand paradox. This illustrates how it is nontrivial to define a 'canonical way' of choosing a random sample from the collection $\mathcal{C}$ of objects of your interest. And in fact, it is impossible to do so without extra structures on $\mathcal{C}$ (such as topological group structure, group action, dynamical system, etc.). Alternatively, on can simply fix a distribution on $\mathcal{C}$ to start with. $\endgroup$ Commented Mar 14, 2023 at 6:57

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I think that you should restrict to bounded convex sets. Finding a natural distribution for them is not easy. The $shape$ of a convex set in the plane is parameterized by a bounded positive measure $\mu$ on $[0,2\pi[$ such that $\int_0^{2\pi}e^{i\theta}\mu(d \theta)=0$ (in a way too long to describe here, which seem to be known only from complex variable specialists) but this is up to a translation, so we have to add a parameter $(t_1,t_2)\in R^2.$ Therefore defining a ransdom convex set in the plane is defining a probability distribution for $(t_1,t_2,\mu).$ Try something more modest, like a random convex polygon inscribed in a given circle...

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